So, it's not an either/or, it can be a both/and.
A really popular way of describing all this is called the Wesleyan Quadrilateral. According to Wesleyan Quadrilateral there are four resources we make appeals to to reach theological conclusions. These are Scripture, Tradition, Reason, and Experience. By Scripture we mean the Bible itself. By Tradition we mean things like the historical creeds which guide readings of Scripture, separating the orthodox from the heterodox. By Reason we mean logical and rational modes of thinking and arguing that monitor how our understandings of God "fit together" with each other, Biblical texts, and the greater tradition. Faith seeking understanding. And finally, there is human experience, how our understanding of God intersects with lived experience, with ourselves, each other, and the world.
Depending upon your religious tradition, the four corners of the Wesleyan Quadrilateral get weighted differently. Painting with a very broad brush, more conservative churches emphasize Scripture and Tradition. More liberal and progressive traditions emphasize Reason and Experience. An example here would be the LGBT community, how literal readings of Scripture from more conservative churches are pitted against the experiences of LGBT persons in progressive churches. You also see this conflict play out in debates about gender roles, where Scriptural texts about female leadership are pitting against the experiences of women in leadership roles in modern society. My interest here isn't to adjudicate any of this for you, simply to draw attention to how Scripture and Experience push and pull against each other in determining how we think about God.
Stepping back and taking a look at the Wesleyan Quadrilateral, it seems odd to think that existential theology wouldn't be a huge resource for us in talking about God. Again, ponder our Augustinian restlessness without God. That restlessness is mostly existential. Without God we feel adrift and lost. Meaning feels fragile. We experience angst and anomie. In Hunting Magic Eels I call it the Ache. From an evangelistic perspective, why wouldn't we want to start a conversation about God with this existential restlessness? And if so, existential theology is a great roadmap of this terrain.
For example, let's go back to Paul Tillich and his essay that appeared in the Saturday Evening Post. In that essay Tillich describes how modern life has become shallow and superficial. Life has lost, in the words of Tillich, the "dimension of depth." And by depth Tillich means existential weight and significance. How did we lose this dimension of depth? Tillich writes:
How did the dimension of depth become lost? ... The loss of the dimension of depth is caused by the relation of man to his world and to himself in our period, the period in which nature is being subjected scientifically and technically to the control of man. In this period, life in the dimension of depth is replaced by life in the horizontal dimension. The driving forces of the industrial society of which we are a part go ahead horizontally and not vertically...
Almost every action of ours has a point; we're trying to get to work, or to find a place to buy a bottle of milk after hours. But we can stop and ask why we're doing these things, and that points us beyond to the significance of these significances. The issue may arise for us in a crisis, where we feel that what has been orienting our life up to now lacks real value, weight...A crucial feature of the malaise of immanence is the sense that all these answers are fragile, or uncertain; that a moment may come, where we no longer feel that our chosen path is compelling, or cannot justify it to ourselves or others. There is a fragility of meaning...[T]he quotidian is emptied of deeper resonance, is dry, flat; the things which surround us are dead, ugly, empty; and the way we organize them, shape them, in order to live has no meaning, beauty, depth, sense...[We now experience] a terrible flatness in the everyday.